In this question Vector bundles on elliptic curves the author states that for $F$ a stable vector bundle of degree $d$ and rank $r$, with $(r,d)$ coprime and $X$ an elliptic curve one can construct an extension $$0 \to H^0(F) \otimes O_X \to G \to F \to 0 $$ such that the boundary map of the associated long sequence in cohomology is given by $Id: H^0(F) \to H^0(F)$. Furthermore, G is a vector bundle of rank $r+d$ and degree $d$.
I do not see why this is true.
I do know that showing $G$ exists is the same as showing that $H^1(F^{\lor} \otimes H^0(F) \otimes \mathcal{O}_X)$ is non-trivial.